Six-Functor Formalisms (Seminar)

1Purpose

The Six functor formalism is a way to formalize the notion of a cohomology theory and allows one to simplify and streamline the proofs of the basic properties. Mann has recently written up notes [Heyer–Mann 2024] regarding these developments and its application in representation theory. In this seminar, we hope to cover some interesting topics about the concept of six-functor formalism and write a readable notes of our own.

2Location and Recordings

Online and Recordings will be uploaded to Bilibili and Youtube.

QQ group : 786106396.

3Note and Schedule(no fixed)

注 3.1. If the content cannot be completed in one talk, it will be postponed to the next talk.

Note(Tex) : 讲义: 六函子理论

(Talk 1)	Introduction and Basic Notion of $\infty$-categories 1. Liu Ou Abstract: ∘ Introducing Six-Functor Formalism. ∘ Why $\mathsf{Corr}$(Construction and proper base change by correspondence). ∘ Why $\infty$-category(Adjoint Functor Theorem). ∘ Two approaches of $(\infty ,1)$-category(Kan-enriched category and quasicategory). ∘ A brief introduction of Model category(Example: Quillen Model Structure, Kan-Quillen Model Structure, Projective Model Structure and Model Structure of $\mathcal{A}$ where $\mathcal{A}$ is Grothendieck Abelian Category, Bergner Model Structure and Joyal Model Structure). Ref: [Heyer–Mann 2024, Appendix A], [Lurie 2018], [Lurie 2009] Time : 12 Jan. Recording : Bilibili(Chinese).
(Talk 2)	Basic Notion of $\infty$-categories 2. Liu Ou Abstract: ∘ Realization-Nerve Adjunction and examples. ∘ More simplicial sets: face morphism and degenerate morphism, dimension of simplicial set, skeleton and coskeleton, spine. ∘ Small object argument, (Trivial Kan/ Kan/ Inner/ Left/ Right) Fibrations and some consequences of Joyal Lifting Theorem. ∘ Introduce Grothendieck-Lurie Construction(definition). Ref: [Heyer–Mann 2024, Appendix A], [Lurie 2018], [Lurie 2009] Time : 19 Jan. Recording : Bilibili(Chinese). Note(Handwriting) : Here.
(Talk 3)	Basic Notion of $\infty$-categories 3. Liu Ou Abstract: ∘ Grothendieck-Lurie Construction(Locally Cocartesian fibration, the construction of straightening theorem and some properties) ∘ Yoneda Lemma. ∘ Adjunction and Quillen Adjunction. ∘ Limits and homotopy limits. Time : 9 Feb. Recording(It didn’t work out so let’s try it again) : Bilibili(Chinese). New Time: 13 Feb Recording : Bilibili(Chinese). Note(Handwriting) : Here.
(Talk 4)	Basic Notion of $\infty$-categories 4. Liu Ou Abstract: ∘ Quillen’s Theorem A. ∘ Homotopy Group Time : 11 Mar. Recording : Bilibili(Chinese).
(Talk 5)	Example — The Six Functor Formalisms of $\mathsf{Cond}(\mathsf{Ani})$. Liu Ou Abstract: ∘ Ind-Completion, Pro-Completion; ∘ Definition of Condensed Animae; ∘ The Six-Functor Formalisms of Condensed Animae. Have some typo. Time: 29 Mar. Recording : Bilibili(Chinese).

4References

•	Peter Scholze (2022). “Six-Functor Formalisms”. (web)
•	Claudius Heyer, Lucas Mann (2024). “6-Functor Formalisms and Smooth Representations”. arXiv: 2410.13038 [math.CT]. (web)
•	Lucas Mann (2022). “A $p$-Adic 6-Functor Formalism in Rigid-Analytic Geometry”. arXiv: 2206.02022 [math.AG]. (web)
•	Lucas Mann (2022). “The 6-Functor Formalism for ${\mathbb{Z}}_{\ell }$- and ${\mathbb{Q}}_{\ell }$-Sheaves on Diamonds”. arXiv: 2209.08135 [math.AG]. (web)
•	Yifeng Liu, Weizhe Zheng (2024). “Enhanced six operations and base change theorem for higher Artin stacks”. arXiv: 1211.5948 [math.AG]. (web)
•	Algebraic Pattern: Shaul Barkan, Rune Haugseng, Jan Steinebrunner (2024). “Envelopes for Algebraic Patterns”. arXiv: 2208.07183 [math.CT]. (web)
•	Enriched $\infty$-categories: Hongyi Chu, Rune Haugseng (2023). “Enriched homotopy-coherent structures”. arXiv: 2308.11502 [math.CT]. (web)
•	Jacob Lurie (2009). Higher Topos Theory. Princeton University Press.
•	Jacob Lurie (2017). Higher Algebra. preprint.
•	Jacob Lurie (2018). Kerodon.
•	Some concepts in 高阶范畴论 and 高阶代数.